Discussions, incl. FW Lawvere, G Khatcherian & M Wright (contd.)

5:00 My own naive way, I thought that super algebras are the onion bhaji, one onion bhaji, one lamb tikka, one onion bhaji and one avocado prawn. Well, that's ok, so we'll share the main course. Among the favourites of commoners is chicken tikka masala. Was it one lamb tikka?

7:30 One lamb tikka. Do you want to have chicken tikka? I'll tell you what, okay, so one curry, one masala, and I'll have the shashlik, and then we've got chicken shashlik, that's right, and then we've got chicken shashlik, yeah, chicken shashlik, yeah, sorry, yes please, what do you recommend? One vegetable curry, I think maybe one mushroom bhaji. Is that okay? Yeah, we'll just have some flour rice. Two please. Okay. Yeah, one half bread. Jokes can be about two-edged weapons though. Sometimes they can diffuse, and I always think of Charlie Chaplin and the Great Dictator, Charlie Chaplin. And also it's set out to be a caricature of Hitler, but it's always one of the most rotten corrupt films ever made, because it's humanised. To make it. Kind of sympathetic comic. Because you can't make something comic. ... and ridiculous without making it in some way sympathetic, in one piece in some way. I mean, I don't think that you can make a totally evil character a comic character. And I think that there is... I mean, I'm aware in myself that I have probably rather a developed sense of humor, I'm not going to pursue development. I'm sometimes aware that it does actually distort my serious... In particular, you know, judgments of politics and events. I mean, I would hate to be taken as dimwit. You'd go for the world, you know. Not being able to see, but taking anything, I mean, that's... Well, I hope there is some... it is something that you have to guard against. It can be very scary.

10:00 Well, classic instance, that. Nothing but that red zone, I think. To try and reduce serious subjects... The subject is demolishing of science to the level of caricature and particularly surrealism. We treat everything as a subject for distortion. Let's take a particular example. We're working on a campus. We're trying to create a kind of mechanical system. That's what I said. Any kind of mechanical system could be used as a system. So there are four categories here. We've got a very grandiose class. 200 original... This is what they're telling us. 200 research mathematicians. Category, very good number. And very good, too. Academicians. Just academicians. Two hundred. And that's, and that's, I mean, just one. Who are the academicians and the academicics? I don't know. If these people were any, any criteria to judge by, you're certainly very, very, uh, substantial. Eh, maybe by dropping the A, you know, after rising. You never saw me. You can't do this, sir. They can't do this, sir. The professor... They can't do this. They can't do this, sir. They can't do this, sir.

12:30 They can't do this, sir. They can't do this, sir. They can't do this, sir. They can't do this, sir. They can't do this, sir. They can't do this, sir. They can't do this, sir. They can't do this, sir. In a monastery, sorry. Do you like that one? He's got the right hardware and the right software. He's a nice fellow because you can even joke about his ideas. Do you like that one? It was spontaneous. That's very good, actually. So was his little cousin. It's a sweet guy. What happened to seminaries? I think so. My memory is... Hang on, the seminary was born in Gorin. It was a seminary in... What was it in Belize? When he rebelled against Leonardo. My memory... Where are we? They were at Pinellas. Oh, Paris. It's a spell for reading Darwin. Darwin and Victor Hugo. And about two dozen other books that were on MediaTek. Thank you for your attention. And both of which, Mikhail Yanovsky. Autonautics, so it's an acronym, you know, the good Soviet, you know, all the way down. Now, the categorical presentation of Autonautics, that's a good one and a half times, it's comfortable. But it didn't work, it was bad, because Krelpo didn't come along. Projectile, I like. Because it fits in with the very geometric way of thinking of all these things. It's too near to the word projectile.

15:00 Projectile? I always call it projectoids. No, no, no, definitely not. Why not? I don't like coins. Projectile, I think. Is it worth the name? For the concert, maybe Bill will tell me. Yes, I am free. I think they're doing a different work. Monod is for you. Anyway, Americans, we have no sense about the monod. They say monod. What does he think? Oh, yes, monod. Yes, yes, it's something bad. Somebody not quite achieved. Schizoid, paranoid, schizoid, and mongoloid, these are the oids that Americans know about. I actually taught Locan, because I mean I'm over the snow. Whether it's worth the name it is all I know, but on the paper it becomes a little bit more of a synchronization. Well, sure, once you've defined it, then by all means use the terms that I've provided you for a dramatic table. E being the neutral one, so I said here, I can't even say it projectively, I think it depends on how I think it is. For a moment or something, it's kind of like an old projection. So I can't even say it projectively. So I said here, and the other one, which is the old one. Yeah, projective mathematics. I'm not all for mathematics, really. The only thing about P.R.E.L., the only thing about P.R.E.L. that when people hear it for the first time, well, obviously if they studied the definition, then they know the definition, and there isn't a problem. But if they haven't studied the definition, I think that a lot of them will think that it's an acronym, that it's P.R.E.L., and they'll probably think the R.E. stands for Right Exact. And they'll get hopelessly confused as to what the concept is, that's why I don't think you can pick it up from the pro.

17:30 Yeah, if you come up with something, it's a PR name. But I like projectile, and I certainly like O'Mickel, I think O'Mickel is a good name for that. I don't know, I mean, it's his name, it's 1887 behind. Well, what did you write about him? He not being here, what would you call him? He's called it subtraction, which is good enough. He actually wrote also the division of some applications. Division? That's a vicious thing. There's a general formulation due to Grasbaum. I think maybe it goes back to that. See, some operations are synthetic, and then you try to undo this, and that's analytics, or addition, but no, in fact, the only thing, they didn't have the notion of analytics. So it's somehow vaguely an engine, but of course it isn't really an engine. It's not a vaccine, it's an analytics. I have a good word for it. You can analyze a synthetic operation when you have it. So the theory now has to do with rotation. Instead of writing B minus A, write the same way as you write A and plus B. Minus A plus B. No, A something B. And call it the A deduct of B. The A deduct of B. The thing you mentioned in your lecture, the implication of deductions. Thank you very much for your attention. Well, that's why I think a little is, exactly why I think a little is a good word, because it's down to the end of the day. But it gets in the spatial thing, it gets in the spatial thing.

20:00 It gets in the region, dealing with the region. I went all out. I went all out and said, I am pleased to be here. I should call it. Topology, geometry, algebra, analysis, quantum mechanics, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, Okay, well, let's be sure we understand the thing. No, no, we understand it. That's what we're talking about here. We understand it. Yeah, I can tell you that. I can tell you that. I can tell you that. But, because of the underlying deduction, take the zero for that, saying the system does not exist. They take one, I mean, the system does exist. And they, you know, the system can read that data. Relativity is a purely theoretical statement. It's where lattice theory, which I think is newly discovered by Yeager, and then responded to by Lansdowne, which is a very controversial statement. Contamination is the kind of statement that is used in mathematics. I think it's very curious that all my students,

22:30 but they also, in therapy I think, have a particular point of view on how this model works. There are many different types of mathematics, but only a few of them are known to the general public. There are many different types of mathematics, but only a few of them are known to the general public. There are many different types of mathematics, but only a few of them are known to the general public. There are many different types of mathematics, but only a few of them are known to the general public.

25:00 There are many different types of mathematics, but only a few of them are known to the general public. Some really off-diagonal elements. I told you this example. I like very simple examples, right? So I was very pleased when I was visiting Greece five years ago, waiting for a bus. Did I tell you this story about waiting for a bus? And the point is, you see, that these people who think that there's only process, no states, that morphisms exist, that they don't have any domains that put them in. They're very much against the stasis, you see. Any ideas for stasis? Dynamics is part of logic. That's actually a Finkelstein. So waiting for the bus, you see, I suddenly realized there's a sign there, you see, that says stasis. Because in modern Greek, the word stasis means bus stop. Oh, I see. I mean, it might mean other things. It means the stop of the bus. Now, you see, this becomes clear then. The bus system, the main thing about the bus system is it goes from place to place. But, if it didn't stop, it wouldn't be... It would be a very social beauty, I'd say. So it's a graph, it's a reflexive graph. Not the main aspect of the system, but it's a necessary aspect. I find this example very compelling. Yeah, I agree. It's a very compelling example. I'm just a little bit worried that journalists become a bit prone to this way of thinking. Well, I've talked to him about local set theory, Thomas's, and this business in his book about reference frame, thinking of truth values in relation to a reference frame, and the fact that he takes up this work of Davis on...

27:30 Think of the reals as self-adjointed operators, which was applied in quantum logic in every, you know, the quantization just is a Boolean valuation, and that it all goes with this idea of indefinite, you know, there being indefinite objects, you know, that John has sort of fallen into this way of thinking, and I think he's, maybe I'm being unfair, maybe I haven't understood it clearly enough. But this seems to be the way that he's, this seems to be the way that he thinks of proposes. The man on your right might know. Have you read John's book? Only very briefly. I just saw it when I got to Milan a couple of times. I read briefly that second. I read the Pal and the Party's review of it. Well, it's not only the book on topos, it's the article on infinitesimals, a very good, very interesting article, and it does end with remarks very much in this vein, and thus we see that the stasis of mathematical entities is... Oh, what is it? Yes, you know, all these remarks about the, we model variation in terms of the static infinites, i.e., and set theory, but the thing about topos is that it hasn't been built, the concept of variation. Well, sure, but therefore we've eliminated stasis altogether, which means we've eliminated... He actually used the word stasis, didn't he? In more than one place, yes. That's right. In the infinitesimals paper. As I said, this was several years ago. My friends in Milan are very fond of this. This becomes a slogan for them. Mavir is for stasis. I don't know what it means. This Greek bus stop. It really means it. Is this thinking dialect? It means it. You mean the buses shouldn't stop in here?

30:00 There are no buses really, they're off. But you see, it's all this, as you say, anti-consciousness, anti-consciousness rejection of experience. This idea that all is flux is, you know, there are no beings. This is a world without beings. There is no category of being. And hence no buses either. Buses are all an illusion, like in Buddhism. I am the batsman and the bat, I am the bowler and the ball, we umpire the pavilion of cats. The Indian guy, you know, he was hurt with his three questions. Oh God, him. Lally, Lally, Lally. Lally, Lally, Lally. No, Bill was far too patient, I think. Bill was painstakingly patient with him. Well, what could he say? Well, an overwhelming desire to tell him to shut up. Well, no, I missed him in the other one. No, but there are, there are. He didn't contribute to the scientific discussion. No, I mean... Yeah. No, but I'm the last person in the world to be elitist. Yeah. Which meant nothing. Yeah, nothing whatever. No, but I mean, obviously, I'm the last person in the world to be in a position to be elitist about anything, but there are certain conventions that should be observed when you've got some of the leading category theorists in Europe, in the world, when you've got film and theatre lecturing to them about the foundations of their subject. I mean, you know, you don't get somebody, a total amateur, absolute outsider, you know, biologist, who starts off the discussion with a five, you know, five minute question and then follows up with three supplementary questions, this one, yes, this one here, yes, this one here. That's true. Well, it's a little bit like the British establishment, I think, between Lenin and Stalin. Remember that? I don't think that the times have been true. No, no, in all of this, you're a show-caller. Good job. Lenin, Stalin. Any more in which every respectful British. Well, I think it's okay, sir.

32:30 Um, can I have my new cigarette, sir? Well, that's okay. The, uh... It's so wonderful, isn't it? The mentality! You know, Michael, I've got cases and cases on my maths books, which are in John Baird's paper. Oh, what's going to happen to them when he sells the house? I'm going to take them away, but I'm wondering where to store them. Well, if you want to store any of them here, you're more than welcome. Particularly since I'm going to be moving them. I've got about two and a half thousand books in the garage at the moment, which has got to be in... Because I've just had a whole lot of bookcases made, and they're just going to be moved into the room where Harvey's sleeping. Right. I mean, use them, but as long as... Well, obviously I'll take care of them for you. I'll make some tea, shall I? Tea or coffee? I haven't got any more wine. I thought, I'm sure you should have got a bottle, but I haven't. They seem to have some connection with my thoughts. You know how to do something you can think about? There's a scene outside the window and the artist draws on a piece of paper and he rolls up the paper and puts it in the tube and mails it to somebody else. In other words, the scene is mapped onto the artist's painting, the painting is mapped into the tube and then the tube is mailed somewhere else. All this mapping is taking place without ever being in the elements. You don't need to have points in the scene, you don't need to have points on the paper.

35:00 It's a queer idea of mathematics. Yeah, that's interesting. In other words, the idea of logarithmic that I tried to offer there in Cambridge was that you want to have the algebraic structure of the independent quantities. What these independent quantities do is record where other more interesting quantities live. So, for example, there's production of wheat in the state right there, so the amount of wheat that's produced is an extensive quantity, but you can say, well, where is wheat produced, so it's in a certain region, those counties, those areas, so there's where there's some wheat, you know, and elsewhere there's no wheat, so you have this region, again, no need of points here, there's this region where... So you can say, well, where is wheat produced and where is cows produced, you know, intersection, union, and all this, but it's, I mean, again, the cows are actual quantities, you know, some are big and some are small, and more cows or less cows, more milk, less milk, and all this, so those are actual extensive and intensive quantities, but in particular, you have sort of degenerate quantities, which are, where are these other quantities, zero or non-zero, more or less, you know. And those define regions, if you wish, and the calculus of those kind of quantities is logic. Is that okay? So it's derived from the geometry, space, intensive and extensive quantity varying over space, motions in space, by passing to this simplified approximation of where does it exist? Where is it zero? What's its support? And then among those things there is a whole calculus, substitution, multiplication, addition. And the assumption that it's the way it supports split. Yes, in the kitchen. It supports in the sense of measured theory or distribution. Yeah, yeah. But don't you also mean it supports that in the sense of a topos, the way that it splits and supports the condition that supports split?

37:30 The support of an object is a sub-object of one, right? Yes, but aren't you thinking of even that in a kind of geometrical sense? Sure. That's right. Those are those kind of quantities, which is just the object itself. The measure is the object itself. In particular, you have, well, where does this object live? That's a sub-object of one. So you pass to the sub-objects aligned by asking, what's the extent of the support of a more interesting object? Yeah, that's what I meant. I mean, you obviously put it much better than I can. Well, that's what I was driving at, yes. It's not connected to splitting, though, is it? No, but... That notion of support, of which one says it might split, that one is a special case where you have these rather objective sorts of quantities of Galileo, Steiner... Cantor, Burnside, Godendie, Sorensen. What is intensive? It is connected with conditioning, I mean, how the way that variation, actual variation... Intensives are ratios of extensives. This is the, well, what Mayberry taught... It says that Newton said in his lectures... Descartes and Newton's equation lecture to the empty room... Extensive, yeah, I know that. If you have two extensive funds, like, say, volume and mass... Then they might have a ratio, which would be density. And that's what density is. In fact, shall I find the quotation from Newton? Yeah, I've got it. It's in Mabry's paper. Sorry? Yeah. The topos are specialized in learning a lot of different things. Oh my god, this is awesome.

40:00 We make the language out of the whole damn topos. And not just the topos, because the topos, the math is called a constant. It appears that the constants you apply to logical operations, the and, the or, quantifier, and so on, build up this duplication on top of these constants that in fact only give you Well, I mean, it's sometimes useful mathematical instruction, but to give it a fundamental significance, I mean, they gather it in a way that's moving in order. So you take all the elements and generate the three groups, and then you come back to the group? Except that instead of something simple like youth operations, we have all the operations of logic, with all the quantifiers and all the things. So it's all about reality. I think so. Yeah, really. But he even does it in a way that won't work. What's your customization? I mean, they're saved and they're safe in case they die now. They say themselves by taking diagrams and having 100 million pages of logical things. No, I don't. I don't mind. We have all this logical symbolism, but these are not diagrams. Look here, look here. Given the purpose, the local language has as the down-type symbols The objects of E other than 1 and omega... Say that again, I don't even know how to pronounce it. No. Given a coco, the local language of E has a ground type symbol. Why do you put them in such strange ways? They match the objects of E other than 1 and omega.

42:30 This is not even coherent, you see. Why, why other than 1 and omega? No, it's not even coherent because if you have a function you need to define, then... It might get confusing. Some objects, if you've not already, they might get identified with one another. And this is just crazy, because you've got a skewed two objects in some construction. It's confusing enough. I hadn't realized we both have. Those are just the ground types. And you have to start over again and build up these formal types on top of those. Plus the logical operations on those things. It's a total dissipation. Just to avoid, just to avoid some bookkeeping, just to avoid reading the guidelines, I think that's the knowledge we're going to need. It's really, you know, when you get right down to it, I mean, the Georgians are not any more crazy than you, you know, because of the volume. Not in my own naive way. The Georgians are not that much more crazy than you. You see, I really, you know, this is what I want to understand more than anything else, how do you really think about logic and geometry and stuff, you know, that's probably the most important. In the summary, you know, what would the realisation be? Yeah, I mean, I, I, I... So what's that? There are regions of the space where quantities work, and you take in themselves the quantities, and then... So you can think of, you know, the existential quantifier that takes the, say, the set of, the sheet... Yeah, so you actually think of a mapping from the set of all sheep that are owned by some man to that man, you know, for all, you know, there exists a sheep that is owned by some man, there exists a man who is the owner of that sheep, there's quite literally a mapping in the same sense that Jerry's thinking of mapping, that's coming from, you know, the very notion of mapping is coming out of our experience with our experience of motion. And you project that into this all in there. And then the image of that is the seaport.

45:00 Like, the slave on it is the slave, right? And you project it into the slave on it. But you see, it's a very, quite literally a projection in a very strong geometrical sense. But you have it more precise, you see, because this man, his mind, owns a certain number of slaves. This would be an interesting, I mean, a more detailed point. But the question of whether he's having his slaves or not, if he needs his slaves or not, it's an important point, isn't it? It's an important point. It's an important point. It's an important point. It's an important point. This quantity is, or the sum of the quantities itself, the general figure, but if you ask where does the sum live, where does x plus x live, it's the same as when the word x lives, meaning non-zero. Let's say it's a positive quantity. It's a positive, yeah. So the positive quantity x, 2x, lives in the same infinite space where x lives, no more no less. And that's why I didn't say, because if you want to be yourself, if you add it to yourself, it's not the same. It just takes the aspects of whether you're here or not. And all those doesn't seem to be useful. If you look for a lot of them, it seems to be nothing else. Well, it's just an extrinsic approximation to the real quantity. We have in our mind, you know, well, for example, cigarettes. I'm interested to know where are the cigarettes. So I know that in cigarette shops there are cigarettes, right? This is the other thing, the fact that they have a precise number. But that's not, you know, the first thing that interests me is whether they have them or not.

47:30 Okay. So you've, uh, you've been in this business for a long time. Nothing to add to this. Uh, yes. Nothing to add to this. I'm just trying to find that paper of John's with the quotations in it, if you can use it. No, it's not. It's a thing called what my mom did. I think, um, yeah, that is the... I mean, let's see how, if we could look at it again, so now... The practical basis is down tight for all the objects except one or more. I think this is indicative of a total lack of mathematical care or confidence in all experiments of every single application of strategy, whether it be for intercalatory theory, for group theory, for analysis, for tech theory, and in many other situations. You know this never works. It never works because arbitrarily you do things. This is not funtorial. If you map one purpose to another purpose, which the properties of not being an omega is not very well preserved. So the full conception is described in this way. And even if you were interested in this, as you might be, if you like to take the use, it's sometimes useful to take the use, but you might be interested in that conception if you simply don't take it. You can't take the non-dual elements of the theory, and that's not a tutorial, as you see. You might want to do that in another that doesn't preserve non-dual elements. So the whole construction is non-concurrent, it becomes a, you know, you get yourself into, in other words, a much more complicated book you can copy by avoiding to deal with the basic books you can copy. And these societies are arbitrarily excluding certain elements. I mean, that's why people find ideas about partially defined mathematics so crazy.

50:00 Exactly. Because they're specifically not quantorial. Not quantorial. Their main detail, their main rule in hell. Well, of course. They're all complicated. They're not really there anyway. Because they aren't. There isn't anything. But I assume I've got time left, I should say. Can I clarify my mind something that's bothering you? If you take a group and you don't take the elements into consideration, you must call in a new group for them, not the original group. Some groups are not clear on this, but I mean, obviously, you know. Do you have a ratio in psychology? This is... Ratio in psychology, I... I'm trying still to release myself out of A and not not, A and not not, B and not not, A and not not. Do you think it's a good formula or what? I do, but it's one of the ones that cannot... Well, I mean... Well, it's... Well, not the right thing. Yeah, I... The nearest I came to finding a tool, because here in the corner, it looks like it does, because it's connected to the social model, but if you look in the corner, you can see that the platform is going, and it's trying to go back to you, and then you can see that the platform is moving in this direction. The basic calculation is the one and the same, whether or not I'm in on the policy or not I'm in on the policy or not I'm in on the policy or not I'm in on the policy or not I'm in on the policy or not.

52:30 We searched every textbook, checked all of our keys, and found this one record. Another thing that used to bother me... That's what we talked about this afternoon, wasn't it? Do you want me to get the notes? I don't want repetitions. Mike, if you have questions, you can put them in my mind. That's not true at all, Gary. I was wondering actually whether this sort of approach may clarify what the point of all this is.

55:00 There was a third page of that which I saw lying around somewhere. Those were the convex sets. There are also things like threads and shafts and particles that come out of the ground and you stretch them. I don't know this one. There are also things like threads and shafts and you stretch them. Like a threshold. No, I don't understand. So the point is, you see, let's take the dual picture, the closed set. So the thing is, if you have two closed sets, and if their union is one... And I'll really be using their cores as one. Cores are smaller, you know. What's the nature of a core? Well, a general closed set has got the core, which is the solid part like this, but it also has drops and chaff. It can have little threads and things. And you think the core eliminates this part, and it's only the solid core. So, perhaps, what I'm trying to do is to construct an explanation of the... No, you, I was happy with that explanation. Now, that, that, this is what, that is the explanation. And the dual, it's just, formally it's a dual, but now, if A and B, somehow we have A and B that can draw a shaft, which can be called a D, and B has to draw a D, and somehow, even if you eliminate this stuff from A and eliminate this stuff from D, the mean is still 1. Okay, I'm too happy.

57:30 I mean, this is not a precise explanation, but I think in these terms you can... At this level... ...you can try to get some sort of a fucking explanation. ...which could not have... ...which could not have made this A, B, and D equal one anymore true. You know, destroy it at that point. You, you hooked me up with... I meant to talk to you in motion. No, you hooked me up... You put me in motion in other ways, but you put me at rest in this one point. So this one's not a very good picture to be hung up on. Well, I found it initially, I found it surprising. No, I agree. I put it in very good terms and you picked up the signal. That's good. We're on the same wavelength. I agree it's surprising. It's something that, in fact, this kind of principle can be used to prove a lot of things. This is a kind of dialectical human algebra. You understand that these co-hiding algorithms, these closed sets, are much more complicated than human algebra. Therefore, nothing could be true. But in fact, even Leibniz's law is true. Yeah, that's a good point. In fact, I would want to read it. But other things are true. The fact that it's a two-page paper. Yeah. There are a great many useful identities. For example, the completely arbitrary A. Now, if you studied Lebesgue and Piano and Curve and all that, you know, the closed set is incredibly complicated. It has countable geometric edges and material matrices. But things are true. Every A is already at the beginning and at the best center of its core and its boundary. And it's incredible. In fact, it is true. All these things can be proved.

1:00:00 So you can still maintain it. It's too simplified to keep the most of all of it. Okay, let me try another answer to stuff like this. Take any bound. It's going to expand or contract. The rate of expansion or contract. I mean, you made this point somewhere in relation to the circumference of the circle, didn't you? Yeah, yeah. That's what you meant. Right, right. You more or less can write down the essential formula, you know, in these things without doing any analysis. It's not doing quantization, you know. It's trying to take a ball or anything. If you take any object and magnify it in the area, for example, I'm saying if you accept this, this could also be proved, I mean, if you accept this, then, for example, it would be lambda x for the area, and it's a step, right, so that's what comes to you, whereas if it comes to you, you want to see something else, and that's what comes to you. So therefore, that's in constant time.

1:02:30 The definition of cos is square root of 5 less than 4, because this has 4 cos's, 4 out of 4, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5, square root of 5,

1:05:00 There's a very important article in this 1174 by Shamilov about the length of the potato, and it's written in a very simple letter. What's the date of 1986? 1986. It must have been a story to you. I don't know. We didn't even hear of it for such a long time. It was 86. It could be. Well, no, in this case it was. I don't know where this place is. But basically about the angles, there should be no difficulty for angles in a plane. If you do it correctly, there is a more serious problem for angles in three-dimensional space, so there is a further serious problem to make it a problem. In fact, it's the result that you cannot define angles by some kind of finite decomposition problem.

1:07:30 There is a serious problem in expressions like mass gain, and then there's the UVA thing, so it's a little easier. It's even connected with the Monarch-Curson Paradox, that is, it must be this, isn't it? That's what's on it. In other words, there's this idea that two things could have the same volume if you can cut them into finite-numbered pieces, each piece is isometric, each piece is, you know, a pendulum in the sense that Christmas could really happen. So that's, I mean, two things would have the same volume if that's true. But conversely, if they have the same volume, then you have to put them in a finite number of parts, and so, you know, it just makes it difficult. Even without the, I mean, the Bonacarsi paradox, which is a blatant and even a twice as odd, but that depends on the action of choice. But even without that, it's still a difficulty. You could define two figures in a three-dimensional space to be equivalent. If you can split them into a finite number of parts, then the corresponding parts are like the methods. I think this equivalency, I think, is much more coarse than this, is much more fine than this volume. It means something like this, that not only do they have the same volume, but moreover, certain angles have rational ratios. Something like that. Whereas...

1:10:00 If you have some irrational ratios, you still have the same value, but if you get an information that's not correct, somehow these irrational ratios can create problems for students. I think Shanwell's idea about the length of the potato, actually, the length of this potato, it's the length of the potato. The discussion of size, as I've told you before, the three-dimensional object, the cap, the circle, and the circle is the length. What is the length of the freedom? What is the multiplication of this? If you have a rectangle, a rectangular polyhedron, a parallel of a pentagon, a parallel of a pentagon, then the length is this. And obviously a certain surface area. And also obviously a certain length, because it takes these edges, and then it's the length of these edges. That's part of the size. It's a necessary unit of the size of the two of our heads. And in fact, this polynomial is not just a constant polynomial. It's really a measure. Every one of the terms is distributed in extensive quantities over it.

1:12:30 Again, this is obvious for the volume. The volume is distributed throughout the unit. And also for the surface area. The surface area is distributed over the boundary. It doesn't live inside. The support is just a boundary, but it's distributed into the boundaries, not just the length is distributed along the edges, so the boundary of the boundary, and even the Euler characteristic is distributed, because you can attribute to each of these eight corners, you have one-eighth of the Euler characteristic, but the reason for it, that's a rectangular one, if you take one which is planted... Then the part of the Euler characteristic, well I mean this is a connected object to the Euler characteristic of one, the actual angle of each corner somehow adds up to, well he has a very physical explanation, so you have this physical object, now you want to put, you want to make a wooden box, so you take some pieces of wood, I'm going to show you how to make a rectangular shape and then glue it on the sides and that tells you the area, but you haven't completely covered it because you left some cracks along the edge. You want to fill in these cracks, so you take a dowel, a long wooden dowel, a round piece of wood, and you cut it in certain angles to make sure it exactly fills the cracks. The length of the table is how much of this wooden dowel you need to completely fill in the cracks. Now you still haven't filled in everything, but you still have the corners, you have these little holes at the corners, even if you cut in the molding, you still have the corners left in the middle. You take a ball, say a ball of putty, and you cut it in a certain way to precisely fill up this hole, and another piece to precisely fill up that hole, and you fill just one ball to fill up all of these corners. However, if the thing is not the right angle... You need a bigger fraction of the ball for one that's standing more open there and a smaller fraction of the ball for one that's more closed.

1:15:00 So the total fraction of the ball that you need to plug the holes in each of these eight corners, that's the distribution of the total Euler characteristic. This is angles, the three-dimensional angle. The angle is really not, it shouldn't be thought of as a multiple of pi, but simply a multiple, a fraction of one. What part of a complete turn, what part of a complete ball? So the other characteristic is that the concentrator and its support is just the corners. The support of the length is just the edges and the support of the areas. It's amazing that all that, that's all true, you see. There's a new polynomial, a polynomial of measures now that has all these properties. It's very, uh, Michael, you did this one. Yeah. It's connected with carbon DNA. Yes, sorry, I was actually reading, I was actually thinking about something Jerry was saying earlier, you know, about the, sorry, say again, what's this? You've been more serious than you were. Yeah. No, no, no, it was about the identical problem. The size of a polynomial is actually an extensive quantity. I mean, the size of a polynomial is the total of an extensive quantity. Each term is really distributed over the body. Obviously, the volume is distributed. Yes. And the area is distributed over the surface, and its support, the logical aspect of the second term is really focused on the surface, but still it's distributed. If you have edges, then the length is distributed along the edges, the total length. And even the order characteristics, the corners. And the way that you find the distribution is basically, you imagine actually making out of wood the sides. Thank you very much. So if it's not rectangular, some corners need more cutting and others need less cutting.

1:17:30 The total that you need is one. Yes, exactly. But the distribution tells you the angle. That would be a marvelous way of teaching children at school about all the types of things. Yeah. Or even the length. But now you see what I'm saying is that with this parallel pipette, the length is, because you have these sharp edges, The linear term is concentrated there. You can understand it. But if you have a ball, a smooth ball, then there still is a length. But it's distributed uniformly all over the surface. I mean, the potato is the length. Yeah, the potato is the length. Oh, this is, uh, Shamuel's paper about the length of the potato, yeah. There's a certain total value that it has. There's also a measure that's distributed over the length. If there's some sharp creases in the potato, then there's a lot more there. But in general, it's sort of smooth movement. But I mean, the funny thing is, if you just give this explanation that I've given, it sounds like an incredibly complicated construction that you'd have to... And some crazy person would think, well, you could actually make it out of wood and bell and plucky and, you know, couldn't possibly be a consistent mathematical theory. There must be some flaw in it. That's a big lick. But in fact, this is the unique polynomial that satisfies very simple relations, namely the size of the Cartesian product is the product of the polynomial. That's a beautiful, beautiful theory. But in particular, you see, it contains angle. Because The distribution of these terms will contain the angular information. I mean, what part of the ball is needed for this corner, what part is needed for that corner? Well, also, how much turning is there in an edge? What fraction of a whole complete disk is involved in the return? In the continuous space, it's the curvature measure that's really the whole question. If it's a sharp edge, then it tends to be concentrated there. A sharp edge takes everybody. They're separating flat planes and all kinds of things there. If you get a course starting with Lebesgue measure, you never get to this. It takes a whole year to explain Lebesgue measure.

1:20:00 Lebesgue measure, though, I remember Lebesgue now so rustlingly, it was a completion of something. Lebesgue thought of it. It meant we had done something. Well, it made sense of the partition. Well, when you say completion, that should be a very good fact, isn't it? Any subset of the set of measurements you have included in the measurement, given the measurements you have? As you were saying this afternoon, I mean, it was really a monster-barring exercise, to use Macintosh's terminology, because it was so hung up on one and zero, and the point of... In reading this journal here, which you now know why, why don't you think Mabry's paper is as bad as that? No, no, I've found it interesting. I shouldn't have read it before. It's wrong, but it's... It's a review of Lactatasha's book. Oh, not the one by Quine. By Quine. Oh, yes, I've read this review. It's extraordinarily shallow. He's telling you what's in that book, and it's nothing but precisely this thing that Shannonwell has refuted. Really? Chattywell had, in fact, refuted the whole thesis of fruits and applications. Yeah. Well, for God's sake. Because, you see, at least I haven't read it, but what Klein says is that the whole book is starting with Euler characteristics. That's exactly right. And showing that, well, for nice circumstances, the value is two. Yes, and you find what was wrong with the earlier definition and stretch the concepts, find a new counterexample, stretch, find a guilty level. Oh, of course it's not like that. This is just taking Popperianism. Oh, yes, but in fact it's well known that Lakatos utterly falsified it. That's a couple of steps one leaps to the general definition of whatever characteristic.

1:22:30 Yeah, cool. And in fact one should have because it's just the hidden potent part of the Burnside rig of the category of polyhedra. Yeah. It has to be more, you know, there's a conceptual explanation. Yes. That's the true dialectic that we have in the whole series. Yes, but you see the whole point is that Lakatos couldn't believe in some two examples. Exactly. He thought he would, you know, he thought that, you know, he thought there would always be a change of dominant theory. Of course, yes, I mean it's all pragmatism. Yes, yes. Well, it's more than that. Well, it is pragmatism. It's also profound historical... Pragmatism disguised as dialectics. I mean, it's called dialectical mathematics. Oh, yes. Well, of course, he always claimed that he'd been heavily influenced by Hegel because he hadn't been a Marxist before his... before he became a Reagan anti-communist. Oh, no. I was perhaps looking again at his paper on... Well, if he'd read Hegel, he might have noticed that line quoted by Lenin. There's not only the abstract general, but also the general that contains all of the examples. Well, I don't know how well he had read Hegel. He certainly read Hegel in the period when he was a Marxist. There's a category of polygamy. You do have the Bernstein remuneration. I mean, the negative part of that is the... Well, it wouldn't be a bad project. I mean, you've got more important things to do, but to write a short polemic against Lakotash, pointing out this very point. This was quite illuminating. I never remembered or realized that that was the only thing he dealt with, was actual Euler characteristics. Oh yes, and he only takes that as far as, well, very early point around. If you rely on the fact that people haven't been taught what arithmetic really is, they'll have some very vague feeling about it. It's a mystery, therefore, it's recent science. It's Keras's program. You mystify recent science in order to promote the actionary philosophy. In Lakdosh's case, that really was so nakedly the open aim of the program. It wasn't disguised really in any form in the way that it usually is. I do buy on his own. I think there's one in the kitchen. It was an extraordinary essay. You really should read some of his stuff. And Quine, incidentally, has the most extraordinarily mystical paper about set theory. The volume of tributes is an Akatosh Memorial volume, after you know. The most extraordinary, really just like, you know, Dawling, or everything, you know, there are no physical, there is no physical world at all.

1:25:00 I met Lakatos, actually. You did? I thought you were there. At the same time when I met Marker Bunger and Mario Bunger. Uh-huh. I met Jim Lombeck, who introduced me to Marker Bunger. Ah, the man who thinks you're a Heraklite. And then... And then Mario will introduce me to his friend, Imre Lakikot. You should read some of his anti-communist letters, because they're actually published in the book that he's taking out there, and you really haven't got a place in it, but I'll tell you what it is. There's one in which he writes the denunciation of things, what it's good or bad. He writes that the theory of navigation is not, as it were, linear, but as, you know, it could bring some horror, saying that these wicked, wicked men who had to derail all their energy to go against the destruction of the water and other weapons for freedom, rather than, you know, rather than construct offences against, you know, the evil empire, you know, which I know all about. Look at why we want this. You know, all the practicals that we, all the...

1:27:30 Well, all of that tradition of renegade, yeah. Yeah, I think whoever saw, you know, through him in 1949, he was there. That's right, yes, he was there. I do not know the circumstances. He had some very strange opinions. Of course, he liked to give it out. He was a very astute falsifier, falsifier. In fact, he was quite naked about this. He said that he was not concerned with rational reconstructions. Rational reconstruction of the history was not the essence of the first year of the... Yeah. In this case, spiritualistic... So shall we discuss the... Yeah, shall we discuss the... Yeah, absolutely. Here's the... This is the paper that we talked about the other day. This is what you wrote. So let's... Let's go through this now. Do you want some more? Sorry, I've got things to take about. I think it would help me if... Yeah. They would help me if you went straight right up to physics. It's very clear. There's nothing wrong with the official source. One or two questions to figure out the ideals, you know, if there's something there. But if I could, I'd be very happy to. Yeah, sure. I'd be very happy to, too. Yeah, let me guess. The non-men may be able to do it, but I find it's sitting on the floor every once in a while. Yeah, I don't know. I don't know. I suffer a little bit from my joints. Joints? Because you call me a young man, you see.

1:30:00 Yeah, you are. Well, I'm 43 years old. You're not, are you? No, seriously, all the bullshit aside, I never thought you were 43. No, I'm 42 years old. I thought you were younger than me. It's been 20 years since I lost kids. Oh, very few kids. Yeah, 12 and a half. I have very, very good connections. You go for long walks, don't you? No, I don't take that for granted. But you're actually very good. But you're actually a very good man, I think, because you've got to go strong. Not the way in smoking, huh? Well, yeah, but I think that's wrong. It's the old, old, old... It's the old, old, old... It's the old, old, old... I know, I know, that's why I said it. I know, I know, that's why I said it. It's just to keep you busy while you're smoking. It's just to keep you busy while you're smoking. Well, was this a disappointment to your company? Well, was this a disappointment to your company? I was married to a woman for four years. I convinced her to go without smoking. Yeah?

1:32:30 You can refer to that? No, after talking so much, I said, have cigarettes, because this could be so interesting. They brought a carton of cigarettes, and of course, I said, no, no, no, no, no, but well, please, please take them. You've actually got, you've put me all down there. Yeah, let me try to... What, you want to do it again? Let's just do it slightly differently. Well, it was very clearly used. Let's take a particular example. Here's another one. Here's one that works. Without metaphysics. Without metaphysics. Without metaphysics. So, this is just for... Well... I'm just going to include that into the logic. Yeah. Where we're going to come to.

1:35:00 Not actually when you were talking about mathematics, but you could go another time. Yeah, right. Anyway, there's many different types of intensive quantities that would be an example. So, for example, you get the population of a company. So, this has two basic properties...